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What is the area of the region defined by the equation $x^2+y^2 - 7 = 4y-14x+3$?

Guest Oct 8, 2017
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Let's get this equation of a circle into the form  (x - h)2 + (y - k)2  =  r2   , where  (h, k)  is the center of the circle, and  r  is its radius.

 

x2 + y2 - 7  =  4y - 14x + 3

                                                  Add  7  to both sides.

x2  +  y2  =  4y - 14x + 10

                                                 Add  14x  to both sides, and subtract  4y  from both sides.

x2 + 14x   +  y2 - 4y    =  10

                                                 Add  49   and add  4  to both sides to complete the squares.

 

x2 + 14x + 49  +  y2 - 4y + 4   =   10 + 49 + 4

                                                                         Factor  x2 + 14x + 49  as a perfect square trinomial.

(x + 7)2   +   y2 - 4y + 4  =  63

                                                                         Factor  y2 - 4y + 4  as a perfect square trinomial.

(x + 7)2   +   (y - 2)2  =  63

 

Now that the equation is in this form, we know that..

 

r2  =  63

 

And...

 

area of circle    =    pi * r2    =    pi * 63    =    63pi

hectictar  Oct 8, 2017

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